Which expression is equivalent to ?
For expressions like this, first look for a common factor in the large terms—here, appears in both. Use the distributive property in reverse: rewrite as , then simplify the smaller binomial . This is usually faster and less error‑prone than fully expanding both products and then combining like terms, though expanding and then refactoring is a good backup to check your work.
Hints
Look for a common factor
Focus on the two large terms and . What factor do they both share?
Use the distributive property in reverse
If you have , you can rewrite it as . Try applying this idea with .
Simplify the inside expression carefully
After factoring out , you will get something like inside the parentheses. Combine like terms correctly to simplify it.
Desmos Guide
Graph the original expression
In Desmos, type y = (2x - 3)(x + 7) - x(x + 7) in one line. This is the function you want to match.
Graph each answer choice
On separate lines, type each option as a new function, for example:
y = (x - 7)(x + 3)y = (x + 3)(x + 7)y = x^2 + 10x + 21y = (x + 7)(x - 3)Use the different colors to tell the graphs apart.
Compare the graphs
Zoom out or move the view so you can clearly see all curves. The expression that is equivalent to the original will have a graph that lies exactly on top of the original graph for every -value (the two curves will be indistinguishable). Identify which option behaves this way.
Step-by-step Explanation
Spot and factor the common term
Look at the two big terms in the expression:
Both terms contain the factor . Use the idea :
Simplify inside the brackets
Now simplify the expression inside the brackets:
So the whole expression is multiplied by this simpler binomial .
Write the final equivalent expression and match the choice
Substitute the simplified binomial back into the factored form:
So an equivalent expression is , which corresponds to choice D.